EFFICIENT ANALYSIS OF MEDICAL IMAGE DE-NOISING FOR MRI AND

ULTRASOUND IMAGES

MOHAMED SALEH ABUAZOUM

A dissertation submitted in fulfillment of the requirement for the award of the

Degree Master of Electrical Engineering

Faculty of Electrical and Electronic Engineering

Universiti Tun Hussein Onn Malaysia

January 2012

v

ABSTRACT

Magnetic resonance imaging (MRI) and ultrasound images have been widely exploited

for more truthful pathological changes as well as diagnosis. However, they suffer from a

number of shortcomings and these includes: acquisition noise from the equipment,

ambient noise from the environment, the presence of background tissue, other organs

and anatomical influences such as body fat, and breathing motion. Therefore, noise

reduction is very important, as various types of noise generated limits the effectiveness

of medical image diagnosis. In this study, an efficient analysis of MRI and ultrasound

modalities is performed. Three experiments have been carried out that include various

filters (Median, Gaussian and Wiener filter) and evaluating the outcomes of medical

image de-noising after applying these three filters by calculating the peak signal-to-

noise ratio (PSNR), which shows that Gaussian filter is better than Median and Wiener

filter.

vi

TABLE OF CONTENTS

CHAPTER TITLE PAGE

DISSERTATION STATUS CONFIRMATION

SUPERVISOR’S DECLARATION

TITLE PAGE i

DECLARATION ii

DEDICATION iii

ACKNOWLEDGEMENT iv

ABSTRACT v

TABLE OF CONTENTS vi

LIST OF TABLES x

LIST OF FIGURES xi

I INTRODUCTION 1

1.1 Background 1

1.2 Problem Statements 2

1.3 Objectives 3

1.4 Scopes 3

vi

CHAPTER TITLE PAGE

II LITERATURE REVIEW 4

2.1 Related Works 4

2.1.1 Ultrasound de-noising 4

2.1.2 MRI de-noising 5

2.2 Medical imaging 6

2.3 Comparison between MRI and ultrasound imaging 8

2.4 Image noise 9

2.4.1 Amplifier Noise (Gaussian Noise) 9

2.4.2 Salt-and-pepper Noise 10

2.4.3 Speckle Noise 10

2.5 Classification of de-noising filters 11

2.6 De-noising filters review 12

2.6.1 Median filter 12

2.6.2 Gaussian filter 12

2.6.3 Wiener filter 13

2.7 Peak signal-to-noise ratio (PSNR) 15

III METHODOLOGY 17

3.1 The concept of de-noising filters 17

3.1.1 Convolution 18

3.1.2 How does Median filter work? 19

3.1.3 How does Gaussian filter work? 21

3.1.4 How does Wiener filter work? 25

3.2 PSNR Calculation 29

3.3 Description of Methodology 30

3.4 Median filter implementation 31

3.4.1 Applying Median filter on MRI image 31

3.4.2 Applying Median filter on Ultrasound Image 32

vii

3.5 Gaussian filter implementation 34

3.4.3 Applying Gaussian filter on MRI image 34

3.4.4 Applying Gaussian filter on Ultrasound Image 35

3.6 Wiener filter implementation 36

3.6.1 Applying Wiener filter on MRI image 36

3.6.2 Applying Wiener filter on Ultrasound Image 38

IV RESULTS AND ANALYSIS 40

4.1 Result of Median filter on MRI and ultrasound 40

4.2 Result of Gaussian filter on MRI and ultrasound 42

4.3 Result of Wiener filter on MRI and ultrasound 44

V CONCLUSION AND RECOMMENDATION 49

5.1 Conclusion 49

5.2 Recommendation 50

REFERENCES 51

xi

LIST OF FIGURES

FIGURE NO. TITLE PAGE

2.1 Classification of de-noising filters 11

3.1 (a) Pixels of image (b) Kernel of filter 18

3.2 Sorting neighbourhood values and determine median value 19

3.3 Assumed pixels window represented on MRI image 20

3.4 Movement of the window 3×3 (mask) on the pixels 20

3.5 Sorting the pixels and determining middle pixel value 20

3.6 Pixels window after applying Median filter on all pixels 21

3.7 One dimensional Gaussians 21

3.8 Two dimensional Gaussians 22

3.9 3×3 windows with 𝝈 = 0.849, 𝝈 = 1, 𝝈 = 2 23

3.10 3×3 windows with σ = 0.849 23

3.11 Assumed pixels window represented on MRI image 23

3.12 Movement of the window 3 x 3 (mask) on the pixels 24

3.13 3×3 windows with σ = 0.849, first window of MRI pixels 24

3.14 Pixels window after applying Gaussian filter on all pixels 25

3.15 Assumed pixels window represented on MRI image 26

3.16 Movement of the window 3 x 3 (mask) on the pixels 26

3.17 First window of the pixels 26

3.18 Pixels window after applying Wiener filter on all pixels 29

3.19 Image de-noising chart 30

3.20 (a) MRI image (b) Ultrasound image 31

xii

FIGURE NO. TITLE PAGE

3.21 MRI image 31

3.22 Noisy MRI image 32

3.23 Ultrasound image 33

3.24 Noisy ultrasound image 33

3.25 MRI image 34

3.26 Noisy MRI image 34

3.27 Ultrasound image 35

3.28 Noisy ultrasound image 36

3.29 MRI image 37

3.30 Noisy MRI image 37

3.31 Ultrasound image 38

3.32 Noisy ultrasound image 38

4.1 PSNR results of de-noised MRI image by Median filter with

windows size from (1×1) to (6×6) with best PSNR= 14.01 dB 40

4.2 De-noised MRI image by Median filter 41

4.3 PSNR results of de-noised ultrasound image by Median filter with

windows size from (1×1) to (6×6) with best PSNR= 14.98 dB 41

4.4 De-noised ultrasound image by Median filter 42

4.5 PSNR results of de-noised MRI image by Gaussian

filter with windows size from (1×1) to (6×6),

𝜎 = 0.849, with best PSNR= 15.53 dB 42

4.6 De-noised MRI image by Gaussian filter 43

4.7 PSNR results of de-noised Ultrasound Image by Gaussian

filter with windows size from (1×1) to (6×6),

𝜎 = 0.849, with best PSNR= 17.37 dB 43

4.8 De-noised ultrasound image by Gaussian filter 44

4.9 PSNR results of de-noised MRI image by Wiener filter with

windows size from (1×1) to (6×6), with best PSNR= 15.35 dB 44

xiii

FIGURE NO. TITLE PAGE

4.10 De-noised MRI image by Wiener filter 45

4.11 PSNR results of de-noised Ultrasound Image by Wiener filte with

windows size from (1×1) to (6×6), with best PSNR= 17.27 dB 45

4.12 De-noised ultrasound image by Wiener filter 46

4.13 (a1) Original MRI, (a2) Original ultrasound image,

(b1) Noisy MRI by White Gaussian Noise,

(b2) Noisy ultrasound image by White Gaussian Noise,

(c1) De-noised MRI by Median filter,

(c2) De-noised ultrasound image by Median filter,

(d1) De-noised MRI by Gaussian filter,

(d2) De-noised ultrasound image by Gaussian filter,

(e1) De-noised MRI by Wiener filter,

(e2) De-noised ultrasound image by Wiener filter. 47

x

LIST OF TABLES

TABLE NO. TITLE PAGE

2.1 Summary of previous study 6

2.2 Comparison between MRI and ultrasound image 8

4.1 PSNR results of applying Median, Wiener and Gaussian

filter on three ultrasound images (US1, US2 and US3)

and three MRI images (MRI1, MRI2 and MRI3) 46

1

CHAPTER I

INTRODUCTION

1.1 Background

The influence and impact of digital images on modern society is tremendous, and

image processing is now a critical component in science and technology. The rapid

progress in computerized medical image reconstruction, and the associated

developments in analysis methods and computer-aided diagnosis, has propelled medical

imaging into one of the most important sub-fields in scientific imaging [1].

The arrival of digital medical imaging technologies such as positron emission

tomography (PET), magnetic resonance imaging (MRI), computerized tomography (CT)

and ultrasound Imaging has revolutionized modern medicine [2]. Today, many patients

no longer need to go through invasive and often dangerous procedures to diagnose a

wide variety of illnesses. With the widespread use of digital imaging in medicine today,

the quality of digital medical images becomes an important issue. To achieve the best

possible diagnosis it is important that medical images be sharp, clear, and free of noise

and artifacts. While the technologies for acquiring digital medical images continue to

improve, resulting in images of higher and higher resolution and quality, removing noise

in these digital images remains one of the major challenges in the study of medical

imaging, because they could mask and blur important subtle features in the images,

many proposed de-noising techniques have their own problems. Image de-noising still

2

remains a challenge for researchers because noise removal introduces artifacts and

causes blurring of the images [3].

This project describes different methodologies for noise reduction (or de-noising)

giving an insight as to which algorithm should be used to find the most reliable estimate

of the original image data given its degraded version. Noise modelling in medical

images is greatly affected by capturing instruments, data transmission media, image

quantization and discrete sources of radiation. Different algorithms are used depending

on the noise model. Most of images are assumed to have additive random noise which is

modelled as a white Gaussian noise. Therefore it is required to exterminate a variety of

types of de-noising algorithm for MRI and ultrasound modalities.

1.2 Problem Statements

Medical images such as magnetic resonance imaging (MRI) and ultrasound

images have been widely exploited for more truthful pathological changes as well as

diagnosis. However, they suffer from a number of shortcomings and these includes:

acquisition noise from the equipment, ambient noise from the environment, the presence

of background tissue, other organs and anatomical influences such as body fat, and

breathing motion. Therefore, noise reduction is very important, as various types of noise

generated limits the effectiveness of medical image diagnosis. The amount of the noise

has the tendency of being either relatively high or low. Thus, it could harshly degrade

the image quality and cause some loss of image information details.

3

1.3 Objectives

The major objective of this project is noise reduction for measurable objectives

are as follows:

1. To investigate medical Image for better diagnosis.

2. To implement the different types of de-noising filters.

3. To observe the images after the de-noising.

4. To evaluate the best from de-noising filters.

1.4 Scopes

This project is primarily concerned with The scope of the project is to focus on

noise removal techniques for medical images (MRI) and Ultrasound:

1. Using Matlab Simulink software for modalities analysis.

2. Using Image Processing Toolbox to adding noise and to apply de-noising

algorithms.

3. Using some of de-noising filters with different coefficients and comparing between

the results.

4

CHAPTER II

LITERATURE REVIEW

2.1 Related Works

2.1.1 Ultrasound de-noising

Some of the best known standard de-speckling filters are the methods of Lee ,Frost

and Kuan filter [4], [5]. These filters use the second-order sample statistics within a

minimum mean squared error estimation approach. Another common de-speckling

approach is the homomorphic Wiener filter [6], where the image is first subjected to a

logarithmic transform and then filtered with an adaptive filter for additive Gaussian noise

[7]. Lee filter is based on the approach that if the variance over an area is low, then the

smoothing will be performed. Otherwise, if the variance is high (e.g. near edges),

smoothing will not be performed. Kuan filter [4] is considered to be more superior to the

Lee filter. It does not make approximation on the noise variance within the filter window.

The filter simply models the multiplicative model of speckle into an additive linear form,

but it relies on the equivalent numbers of looks (ENL) from an image to determine a

different weighted W to perform the filtering as shown in Eq (1).

𝑊 = 1 −𝐶𝑢

𝐶𝑖 1 + 𝐶𝑢 (1)

5

Where 𝐶𝑢 is the noise variation coefficient and 𝐶𝑖 is the image variation coefficient. Next,

the Wiener is a low pass filter that filters an intensity image that has been degraded by

constant power additive noise. It uses a pixel wise adaptive Wiener method based on

statistics estimated from a local neighbourhood of each pixel.

2.1.2 MRI de-noising

A multitude of variation methods based on partial differential equations have been

developed for a wide variety of images and applications [8], with some of these have been

applied to MRI [9], [10]. However, such methods impose certain kinds of models on local

image structure that are often too simple to capture the complexity of anatomical MR

images. These methods, typically, does not considered the bias introduced by Rician

noise.

Healy and Weaver [11] were among the first to apply soft-thresholding based on

wavelet techniques for de-noising MR images. Nowak [12], operating on the square

magnitude MRI image, includes a Rician noise model in the threshold-based wavelet de-

noising scheme and thereby corrects for the bias introduced by the noise. Pizurica et al.

[13] rely on the prior knowledge of the correlation of wavelet coefficients that represent

significant features across scales. Table 2.1 shows the previous study of de-noising.

6

Table 2.1: Summary of previous study

Images Algorithms References

Ultrasound Image

Wavelet Thresholding

Novel Bayesian Multiscale

[5]

[7]

Magnetic resonance image

(MRI)

Nonlinear anisotropic

wavelet transforms

[10]

[11]

Computed tomography

(CT)

multi-dimensional adaptive filter

Edge-preserving Adaptive Filters

[14]

[15]

2.2 Medical imaging

Medical imaging is the technique and process used to create images of the human

body (or parts and function thereof) for clinical purposes (medical procedures seeking to

reveal, diagnose or examine disease) or medical science (including the study of normal

anatomy and physiology) [16]. Although imaging of removed organs and tissues can be

performed for medical reasons, such procedures are not usually referred to as medical

imaging. As a discipline and in its widest sense, it is part of biological imaging and

incorporates radiology (in the wider sense), nuclear medicine, investigative radiological

sciences, endoscopy, (medical) thermography, medical photography and microscopy (e.g.

for human pathological investigations). Measurement and recording techniques which are

not primarily designed to produce images, such as electroencephalography (EEG),

magneto encephalography (MEG), Electrocardiography (EKG) and others, but which

produce data susceptible to be represented as maps, can be seen as forms of medical

imaging [17].

Radiation exposure from medical imaging in 2006 made up about 50% of total

ionizing radiation exposure in the United States. In the clinical context, "invisible light"

medical imaging is generally equated to radiology or "clinical imaging" and the medical

practitioner responsible for interpreting (and sometimes acquiring) the images is a

radiologist. "Visible light" medical imaging involves digital video or still pictures that can

7

be seen without special equipment. Dermatology and wound care are two modalities that

utilize visible light imagery. Diagnostic radiography designates the technical aspects of

medical imaging and in particular the acquisition of medical images. The radiographer or

radiologic technologist is usually responsible for acquiring medical images of diagnostic

quality, although some radiological interventions are performed by radiologists. While

radiology is an evaluation of anatomy, nuclear medicine provides functional assessment

[18]. Many of the techniques developed for medical imaging also have scientific and

industrial applications. Medical imaging is often perceived to designate the set of

techniques that non-invasively produce images of the internal aspect of the body. In this

restricted sense, medical imaging can be seen as the solution of mathematical inverse

problems. This means that cause (the properties of living tissue) is inferred from effect (the

observed signal). In the case of ultrasonography the probe consists of ultrasonic pressure

waves and echoes inside the tissue show the internal structure. In the case of projection

radiography, the probe is X-ray radiation which is absorbed at different rates in different

tissue types such as bone, muscle and fat. The term noninvasive is a term based on the fact

that following medical imaging modalities do not penetrate the skin physically. But on the

electromagnetic and radiation level, they are quite invasive. From the high energy photons

in X-Ray Computed Tomography, to the 2+ Tesla coils of an MRI device, these modalities

alter the physical and chemical environment of the body in order to obtain data [19].

8

2.3 Comparison between MRI and ultrasound imaging

Digital medical images involving many types of images which are different from one

to another in terms of how is produced and how it is look. In this study MRI images and

ultrasound images are used and Table 2.2 describes the difference between them.

Table 2.2: Comparison between MRI and Ultrasound

9

2.4 Image noise

Image noise is the random variation of brightness or color information in images

produced by the sensor and circuitry of a scanner or digital camera. Image noise can also

originate in film grain and in the unavoidable shot noise of an ideal photon detector [20].

Image noise is generally regarded as an undesirable by-product of image capture. Although

these unwanted fluctuations became known as "noise" by analogy with unwanted sound

they are inaudible and actually beneficial in some applications, such as dithering. The

characteristics of noise depend on its source. The filter or the operator which best reduces

the effect of noise also depends on the source [21]. Many image-processing packages

contain operators to artificially add noise to an image. Deliberately corrupting an image

with noise allows us to test the resistance of an image-processing operator to noise and

assess the performance of various noise filters.

2.4.1 Amplifier Noise (Gaussian Noise)

The standard model of amplifier noise is additive, Gaussian, independent at each

pixel and independent of the signal intensity.In color cameras where more amplification is

used in the blue color channel than in the green or red channel, there can be more noise in

the blue channel .Amplifier noise is a major part of the "read noise" of an image sensor,

that is, of the constant noise level in dark areas of the image [20]. Gaussian

noise is statistical noise that has its probability density function equal to that of the normal

distribution, which is also known as the Gaussian distribution. In other words, the values

that the noise can take on are Gaussian-distributed. A special case is white Gaussian noise,

in which the values at any pairs of times are statistically independent (and uncorrelated). In

applications, Gaussian noise is most commonly used as additive white noise to

yield additive white Gaussian noise. If the white noise sequence is a Gaussian sequence,

then is called a white Gaussian noise (WGN) sequence [21].

10

2.4.2 Salt-and-pepper Noise

An image containing salt-and-pepper noise will have dark pixels in bright regions

and bright pixels in dark regions [20]. This type of noise can be caused by dead pixels,

analog-to-digital converter errors, bit errors in transmission, etc.This can be eliminated in

large part by using dark frame subtraction and by interpolating around dark/bright pixels.

This noise is named for the salt and pepper appearance an image takes on after being

degraded by this type of noise [21].

2.4.3 Speckle Noise

Speckle noise is a granular noise that inherently exists in and degrades the quality of

the active radar and synthetic aperture radar (SAR) images. Speckle noise in conventional

radar results from random fluctuations in the return signal from an object that is no bigger

than a single image-processing element. It increases the mean grey level of a local area.

Speckle noise is caused by signals from elementary scatterers, the gravity-capillary ripples,

and manifests as a pedestal image, beneath the image of the sea waves. [22].

11

2.5 Classification of De-noising filters

As shown in Figure 2.1, there are two basic approaches to image de-noising, spatial

filtering methods and transform domain filtering methods [23]. A traditional way to

remove noise from image data is to employ spatial filters. Spatial filters can be further

classified into non-linear and linear filters. Filtering operations in the wavelet domain can

be subdivided into linear and nonlinear methods.

Figure 2.1: Classification of De-noising Algorithms

12

2.6 De-noising filters review

2.6.1 Median filter

The median filter is a popular nonlinear digital filtering technique, often used to

remove noise. Such noise reduction is a typical pre-processing step to improve the results

of later processing (for example, edge detection on an image). Median filtering is very

widely used in digital image processing because under certain conditions, it preserves

edges while removing noise [24]. Sometimes known as a rank filter, this spatial filter

suppresses isolated noise by replacing each pixel’s intensity by the median of the

intensities of the pixels in its neighbourhood. It is widely used in de-noising and image

smoothing applications. Median filters exhibit edge-preserving characteristics (cf. linear

methods such as average filtering tends to blur edges), which is very desirable for many

image processing applications as edges contain important information for segmenting,

labelling and preserving detail in images. This filter may be represented by Eq (2).

𝐺 𝑢, 𝑣 = 𝑚𝑒𝑑𝑖𝑎𝑛 𝐼 𝑥, 𝑦 , 𝑥,𝑦 ∈ 𝑤𝐹 (2)

where

𝑤𝐹 = 𝑤 𝑥 𝑤 Filter window with pixel (𝑢, 𝑣) as its middle

2.6.2 Gaussian filter

Gaussian filter is a filter whose impulse response is Gaussian function [25].

Gaussian filters are designed to give no overshoot to a step function input while

minimizing the rise and fall time. This behaviour is closely connected to the fact that the

Gaussian filter has the minimum possible group delay. Mathematically, a Gaussian filter

modifies the input signal by convolution with a Gaussian function; this transformation is

also known as the Weierstrass transform. Smoothing is commonly undertaken using linear

filters such as the Gaussian function (the kernel is based on the normal distribution curve),

which tends to produce good results in reducing the influence of noise with respect to the

13

image. The 1D and 2D Gaussian distributions with standard deviation for a data point (x)

and pixel (x,y), are given by Eq (3) and Eq (4), respectively [26].

𝐺 𝑥 =1

2𝜋𝜎2. 𝑒

−𝑥2

2𝜎2 (3)

𝐺 𝑥,𝑦 =1

2𝜋𝜎2. 𝑒

−𝑥2+𝑦2

2𝜎2 (4)

The kernel could be extended to further dimensions as well. For an image, the 2D

Gaussian distribution is used to provide a point-spread; i.e. blurring over neighbouring

pixels. This is implemented on every pixel in the image using the convolution operation.

The degree of blurring is controlled by the sigma or blurring coefficient, as well as the size

of the kernel used (squares with an odd number of pixels; e.g. 3×3, 5×5 pixels, so that the

pixel being acted upon is in the middle). The processing can be speeded up by

implementing the filtering in the frequency rather than spatial domain, especially for the

slower convolution operation (which is implemented as the faster multiplication operation

in the frequency domain).

2.6.3 Wiener filter

Wiener filters are a class of optimum linear filters which involve linear estimation

of a desired signal sequence from another related sequence. It is not an adaptive filter. The

wiener filter’s main purpose is to reduce the amount of noise present in a image by

comparison with an estimation of the desired noiseless image. The Wiener filter may also

be used for smoothing. This filter is the mean squares error-optimal stationary linear filter

for images degraded by additive noise and blurring. It is usually applied in the frequency

domain (by taking the Fourier transform) [21], due to linear motion or unfocussed optics

Wiener filter is the most important technique for removal of blur in images. From a signal

processing standpoint. Each pixel in a digital representation of the photograph should

represent the intensity of a single stationary point in front of the camera. Unfortunately, if

14

the shutter speed is too slow and the camera is in motion, a given pixel will be an

amalgram of intensities from points along the line of the camera's motion.

The goal of the Wiener filter is to filter out noise that has corrupted a signal. It is

based on a statistical approach. Typical filters are designed for a desired frequency

response. The Wiener filter approaches filtering from a different angle. One is assumed to

have knowledge of the spectral properties of the original signal and the noise, and one

seeks the LTI filter whose output would come as close to the original signal as possible

[27]. Wiener filters are characterized by the following:

1. Assumption: signal and (additive) noise are stationary linear random processes

with known spectral characteristics.

2. Requirement: the filter must be physically realizable, i.e. causal (this requirement

can be dropped, resulting in a non-causal solution).

3. Performance criteria: minimum mean-square error.

Wiener Filter in the Fourier Domain as in Eq (5).

𝐺 𝑢,𝑣 =𝐻∗ 𝑢,𝑣 𝑃𝑠 𝑢,𝑣

⎹𝐻 𝑢,𝑣 ⎹ 2 𝑃𝑠 𝑢, 𝑣 + 𝑃𝑛 𝑢, 𝑣 (5)

Where

𝐻(𝑢, 𝑣) = Fourier transform of the point spread function

𝑃𝑠(𝑢, 𝑣) = Power spectrum of the signal process, obtained by taking the Fourier

transform of the signal autocorrelation

𝑃𝑛(𝑢, 𝑣) = Power spectrum of the noise process, obtained by taking the Fourier

transform of the noise autocorrelation

It should be noted that there are some known limitations to Wiener filters. They are

able to suppress frequency components that have been degraded by noise but do not

reconstruct them. Wiener filters are also unable to undo blurring caused by band limiting

of 𝐻(𝑢, 𝑣), which is a phenomenon in real-world imaging systems.

15

2.7 Peak Signal-to-Noise Ratio (PSNR)

Peak signal-to-noise ratio (PSNR) is the ratio between a signal's maximum power

and the power of the signal's noise. Engineers commonly use the PSNR to measure the

quality of reconstructed images that have been compressed. Each picture element (pixel)

has a colour value that can change when an image is compressed and then uncompressed.

Signals can have a wide dynamic range, so PSNR is usually expressed in decibels, which is

a logarithmic scale [28].

The PSNR is most commonly used to measure quality of reconstruction of lossy

compression codecs (e.g., for image compression). The signal in this case is the original

data, and the noise is the error introduced by compression. When comparing compression

codecs it is used as an approximation to human perception of reconstruction quality,

therefore in some cases one reconstruction may appear to be closer to the original than

another, even though it has a lower PSNR (a higher PSNR would normally indicate that

the reconstruction is of higher quality). One has to be extremely careful with the range of

validity of this metric; it is only conclusively valid when it is used to compare results from

the same codec (or codec type) and same content [29]. It is most easily defined via

the mean squared error (MSE), where it denotes the mean square error for two m×n images

I (i, j)& I (i, j) where one of the images is considered a noisy approximation of the other

and is given by Eq (6) and Eq (7).

𝑀𝑆𝐸 = 1

𝑚𝑛 [𝐼 𝑖, 𝑗 − 𝐾 𝑖, 𝑗 ]2 (6)

𝑛−1

𝑗=0

𝑚−1

𝑖=0

The PSNR is defined by Eq (7)

PSNRdB = 10 ∙ log10 MAXI

2

MSE (7)

16

Where, 𝑀𝐴𝑋𝐼 is the maximum possible pixel value of the image. When the pixels are

represented using 8 bits per sample, which is equal to 255.

Typical values for the PSNR in lossy image and video compression are between 30

and 50 dB, where higher is better. Acceptable values for wireless transmission quality loss

are considered to be about 20 dB to 25 dB [30].

17

CHAPTER III

METHODOLOGY

3.1 The concept of de-noising filters

The idea of every de-noising filter is different from other filters because every

filter has its own function. To give simple explanation of the de-noising filters window

3×3 is used for median, Gaussian and Wiener filter.

Before the beginning of the discretion of de-noising filters we have to understand the

convolution.

3.1.1 Convolution

Convolution is a simple mathematical operation which is fundamental for many

common image processing operators. Convolution provides a way of `multiplying

together' two arrays of numbers, generally of different sizes, but of the same

dimensionality, to produce a third array of numbers of the same dimensionality. This can

be used in image processing to implement operators whose output pixel values are

simple linear combinations of certain input pixel values [31].

In an image processing context, one of the input arrays is normally just a gray level

image. The second array is usually much smaller, and is also two-dimensional (although

it may be just a single pixel thick), and is known as the kernel. Figure 3.1, shows an

example image and kernel that we will use to illustrate convolution.

18

(a) (b)

Figure 3.1: (a) Pixels of image (b) Kernel of filter

The convolution is performed by sliding the kernel over the image, generally starting at

the top left corner, so as to move the kernel through all the positions where the kernel

fits entirely within the boundaries of the image. Each kernel position corresponds to a

single output pixel, the value of which is calculated by multiplying together the kernel

value and the underlying image pixel value for each of the cells in the kernel, and then

adding all these numbers together. So, in our example, the value of the bottom right

pixel in the output image will be given by Eq (8).

𝑂1 = 𝐼1 𝑘11 + 𝐼2 𝐾12 + 𝐼3 𝐾13 + 𝐼11 𝐾21 + 𝐼12 𝐾22 + 𝐼13 𝐾𝐾23 + 𝐼21 𝐾31

+ 𝐼22 𝐾32 + 𝐼23 𝐾33 (8)

If the image has M rows and N columns, and the kernel has m rows and n columns, then

the size of the output image will have M - m + 1 rows, and N - n + 1 columns.

Mathematically we can write the convolution as Eq (9).

𝑂 𝑖, 𝑗 = 𝐼 𝑖 + 𝑘 − 1, 𝑗 + 𝑙 − 1 𝐾(𝑘, 𝑙)

𝑛

𝑙=1

𝑚

𝑘=1

(9)

where i runs from 1 to M - m + 1 and j runs from 1 to N - n + 1.

19

3.1.2 How does Median filter work?

The median filter considers each pixel in the image in turn and looks at its nearby

neighbours to decide whether or not it is representative of its surroundings. Instead of

simply replacing the pixel value with the mean of neighbouring pixel values, it replaces

it with the median of those values [21].

Median filter controls the pepper and Gaussian noises. The median filter is reputed to be

edge preserving. The transfer function used here in Eq (10)

𝑇 𝑥,𝑦 = 𝐼 𝑛 𝑥 𝑛

2 (10)

𝐼1 𝐼2 𝐼3 … 𝐼𝑛 𝑥 𝑛

where 𝐼 (𝑛 𝑥 𝑛) / 2 is the intensity value in the middle position of the sorted array of

the neighbouring pixels.

Neighbourhood values are (0, 0, 0, 0, 0 , 4, 4, 12, 22)

Median value is 0

Figure 3.2: Sorting neighbourhood values and determine median value

The median is calculated by first sorting all the pixel values from the surrounding

neighbourhood into numerical order and then replacing the pixel being considered with

the middle pixel value as was showed in Figure 3.2. If the neighbourhood under

consideration contains an even number of pixels, the average of the two middle pixel

values is used. The pattern of neighbours is called the "window", which slides, pixel by

pixel over the entire image.

20

Figure 3.3: Assumed pixels window represented on MRI image

Median filtering using a 3×3 sampling window with the extending border values outside

with 0s

Figure 3.4: Movement of the window 3×3 (mask) on the pixels

Figure 3.5: Sorting the pixels and determining middle pixel value

21

Figure 3.6: Pixels window after applying Median filter on all pixels

3.1.3 How does Gaussian filter work?

Gaussian filter are a class of low-pass filter, all based on the Gaussian

probability distribution function used to blur images and remove noise and detail. In

one dimension, the Gaussian function is here in Eq (11).

𝐺 𝑥 =1

2𝜋𝜎2. 𝑒

−𝑥2

2𝜎2 (11)

Where 𝜎 is the standard deviation of the distribution The distribution is assumed to have

a mean of 0. Shown graphically, we see the familiar bell shaped Gaussian distribution,

where a large value of 𝜎 produces to a flatter curve, and a small value leads to a

“pointier” curve. Figure 3.7 shows examples of such one dimensional

Gaussians [25].

Large value of 𝜎 Small value of 𝜎

Figure 3.7: One dimensional Gaussians

22

When working with images, we need to use the two dimensional Gaussian function

Figure 3.8. This is simply the product of two 1D Gaussian functions (one for each

direction) and is given by Eq (12).

𝐺 𝑥,𝑦 =1

2𝜋𝜎2. 𝑒

−𝑥2+𝑦2

2𝜎2 (12)

𝜎 = 9 𝜎 = 3

Figure 3.8: Two dimensional Gaussians

The Gaussian filter works by using the 2D distribution as a point-spread function. This is

achieved by convolving the 2D Gaussian distribution function with the image. Before

we can perform the convolution a collection of discrete pixels we need to produce a

discrete approximation to the Gaussian function. In theory, the Gaussian distribution is

non-zero everywhere, which would require an infinitely large convolution kernel, but in

practice it is effectively zero more than about three standard deviations from the mean,

and so we can truncate the kernel at this point. The kernel coefficients diminish with

increasing distance from the kernel’s centre. Central pixels have a higher weighting than

those on the periphery [25]. Larger values of 𝜎 produce a wider peak (greater blurring).

Kernel size must increase with increasing 𝜎 to maintain the Gaussian nature of the

filter. Gaussian kernel coefficients depend on the value of 𝜎. Figure 3.9 shows a

different convolution kernel that approximates a Gaussian with 𝜎.

23

Figure 3.9: 3×3 windows with 𝝈 = 0.849, 𝝈 = 1, 𝝈 = 2

The idea of these windows distribution comes from

1

16× =

Figure 3.10: 3×3 windows with σ = 0.849

Where 𝜎 = 0.849 and 16 is the summation of the values in 3×3 window with 𝜎 =

0.849 this window is using to explain how Gaussian filter is working by convolute it on

the pixels of MRI.

Figure 3.11: Assumed pixels window represented on MRI image

24

Figure 3.12: Movement of the window 3 x 3 (mask) on the pixels

To achieve the function of Gaussian filter, the 2D Gaussian distribution of 3×3 window

with 𝜎 = 0.849 must be convolved on the first window of MRI pixels to get first value

of filtered pixel.

1

16 ×

×

Figure 3.13: 3×3 windows with σ = 0.849, first window of MRI pixels

1𝑠𝑡 𝑃 =1 × 0 + 2 × 0 + 1 × 0 + 2 × 0 + 4 × 12 + 2 × 9 + 1 × 0 + 2 × 22 + 1 × 17

16

1𝑠𝑡 𝑓𝑖𝑙𝑡𝑒𝑟𝑒𝑑 𝑝𝑖𝑥𝑒𝑙 = 1127

16= 7.9375 = 8

51

[1] S. Arivazhagan, S. Deivalakshmi, K. Kannan, B.N.Gajbhiye, C.Muralidhar, Sijo

N Lukose, M.P.Subramanian, “Performance Analysis of Wavelet Filters for Image

Denoising”, Advances in Computational Sciences and Technology, Vol.1 No. 1 (2007),

pp. 1–10, ISSN 0973-6107.

[2] Nguyen Thanh Binh and Ashish Khare, "Adaptive complex wavelet technique for

medical image denoising," in proceedings of the third International Conference on the

development of Biomedical Engineering, pp. 195-198, Vietnam, January 11-14, 2010

[3] Y. Wang and H. Zhou, “Total variation wavelet-based medical image de-noising”

School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0430, USA,

2006.

[4] B. Goossens, A. Pizurica, and W. Philips. Image denoising using mixtures of

projected Gaussian scale mixtures, IEEE Transactions on Image Processing, Vol.18,

Issue. 8, pp.1689-1702, 2009.

[5] S. Sudha,., G. Suresh,. and R Sukanesh. Speckle noise reduction in ultrasound

images using context-based adaptive wavelet thresholding, IETE journal, Vol. 55, pp

135-143, 2009.

[6] S. Solbø and T. Eltoft, “A Stationary Wavelet-Domain Wiener Filter for

Correlated Speckle”, IEEE Transactions on geoscience and remote sensing, vol. 46, NO.

4, APRIL 2008.

[7] A. Achim, and A. Bezerianos. Novel Bayesian Multiscale Method for Speckle

Removal in Medical Ultrasound Images, IEEE Transactions on Medical Imaging, Vol.

20, No. 8,pp.772-783, 2001.

[8] P. Perona and J. Malik. Scale-space and edge detection using anisotropic

diffusion. IEEE Trans. Pattern Anal. Mach. Intell., 12(7):629–639, July 1990.

52

[9] M. Lysaker, A. Lundervold, and X. Tai. Noise removal using fourth-order

partial differential equation with applications to medical magnetic resonance images in

space and time. IEEE Trans. Imag. Proc., 2003.

[10] G. Gerig, O. Kubler, R. Kikinis, and F. A. Jolesz. Nonlinear anisotropic filtering

of MRI data. IEEE Tr. Med. Imaging, 11(2):221-232, 1992.

[11] D. Healy and J. Weaver. Two applications of wavelet transforms in magnetic

resonance imaging. IEEE Trans. Info. Theory, 38(2):840-860, 1992.

[12] R. Nowak. Wavelet-based rician noise removal for magnetic resonance imaging.

IEEE Trans. Image. Proc., 8:1408-1419, 1999.

[13] A. Pizurica, W. Philips, I. Lemahieu, and M. Acheroy. A versatile wavelet

domain noise filtration technique for medical imaging. IEEE Trans. Med. Imaging,

22(3):323-331, 2003.

[14] M. Kachelrie, O. Watzke and W. Generalized. multi-dimensional adaptive

filtering for conventional and spiral single-slice, and cone-beam CT.American

Association of Physicists in Medicine,vol.28, no.4, 475-490, (2001).

[15] M. Okumura, O. Takamasa, and S. Tsukagoshi .New Method of Evaluating Edge-

preserving Adaptive Filters for Computed Tomography (CT) : Digital Phantom Method.

Jpn J Radiol Technol, vol.62, no.7, 971-978, (2006).

[16] B.N.Raasch, E.W. Carsky, E. J. Lane, J. P.O’Callaghan, and E. R. Heitzman,

“Radiographic anatomy of the interlobar fissures: A study of 100 specimens,” Amer. J.

Roentgenol., vol. 138, pp. 647–554, 1982.

53

[17] CA Roobottom, G. Mitchell, Morgan-Hughes. "Radiation-reduction strategies in

cardiac computed tomographic angiography". Clin Radiol 65 (11): 859–67.

doi:10.1016/j.crad.2010.04.021. G (November 2010)

[18] S.M. Larie and S.S. Abukmeil. “Brain abnormality in schizophrenia: a systematic

and quantitative review of volumetric magnetic resonance imaging studies”. J. Psych.,

172:110–120, 1998.

[19] Q. Long, X.Y. Xu, M.W. Collins, M. Bourne, and T.M. Griffith. “Magnetic

resonance image processing and structured grid generation of a human abdominal

bifurcation”. Computer Methods and Programs in Biomedicine, 56:249–259, 1998.

[20] S. Zhong, V.Cherkassky,“Image Denoising using Wavelet Thresholding and

Model Selection”. IEEE Trans Image Process 2000. International Conference on,

Volume: 3, 10-13 Sept. 2000 Pages: 262.

[21] S. Kumar, P. Kumar, M. Gupta and A. K. Nagawat, “Performance Comparison of

Median and Wiener Filter in Image De-noising,” International Journal of Computer

Applications 12(4):27–31, December 2010.

[22] S. Kent, O. N. Oçan, and T. Ensari. "Speckle Reduction of Synthetic Aperture

Radar Images Using Wavelet Filtering". in ITG, VDE, FGAN, DLR, EADS, astrium.

EUSAR 2004 Proceedings, 5th European Conference on Synthetic Aperture Radar, May

25–27, 2004, Ulm, Germany.

[23] M. C. Motwani, M. C. Gadiya, R. C. Motwani and F. C. Harris. “Survey of

Image Denoising Techniques”. University of Nevada, Reno Dept of Comp. Sci. & Engr.,

Reno, NV 89557 USA (775) 784-6571.

54

[24] H. Raymond Chan, Chung-Wa Ho, and M. Nikolova, “Salt-and-Pepper noise

removal by median-type noise detectors and detail-preserving regularization”. IEEE

Trans Image Process 2005; 14: 1479-1485.

[25] P. Hsiao1, S. Chou, and F. Huang, “Generic 2-D Gaussian Smoothing Filter for

Noisy Image Processing”. National University of Kaohsiung, Kaohsiung, Taiwan, 2007.

[26] R. Fisher, S. Perkins, A. Walker, E. Wolfart. Gaussian smoothing, Hypermedia

image processing reference (HIPR2), 2003. Available from:URL:http:// homepages .inf.

ed.ac.uk / rbf/ HIPR2/gsmooth.htm.

[27] Kazubek, “Wavelet domain image de-noising by thresholding and Wiener

filtering”. M. Signal Processing Letters IEEE, Volume: 10, Issue: 11, Nov. 2003 265

Vol.3.

[28] M. Kachelrie, O. Watzke and W. Generalized. multi-dimensional adaptive

filtering for conventional and spiral single-slice, and cone-beam CT.American

Association of Physicists in Medicine,vol.28, no.4, 475-490, (2001).

[29] Q. Huynh-Thu and M. Ghanbari, “Scope of validity of PSNR in image/video

quality assessment”. Electronics Letters 44 (13): 800–801, 2008.

[30] N. Thomos, V. Boulgouris, M. G. Strintzis, “Optimized Transmission of

JPEG2000 Streams Over Wireless Channels”. IEEE Transactions on Image Processing ,

15 (1)2006.

[31] R Fisher, S Perkins, A Walker, E Wolfart. Convolution, Hypermedia image

processing reference (HIPR2), 2003. Available from:URL:http:// homepages .inf.

ed.ac.uk / rbf/ HIPR2/convolution.htm.

55

[32] H. Yoshino, C. Dong, Y. Washizawa and Y. Yamashita. “Kernel Wiener Filter and

its Application to Pattern Recognition”, IEEE Transactions on neural networks, 2010.

[33] T. Wiegand, H. Schwarz, A. Joch, F. Kossentini, and G. Sullivan, “Rate-

constrained coder control and comparison of video coding standards”, IEEE Trans.

Circuits Syst. Video Technol., 2003, 13, (7), pp. 688–703